Cyclic Steady-State Simulation and Waveform Design for Dynamic/Programmable Catalysis
Carolina Colombo Tedesco, John R. Kitchin, Carl D. Laird, “Cyclic Steady-State Simulation and Waveform Design for Dynamic/Programmable Catalysis”, The Journal of Physical Chemistry C 128(22), 8993–9002 (2024). https://doi.org/10.1021/acs.jpcc.4c01543
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Note on the source text: the extracted text is the open-access (CC-BY 4.0) article, part of the Jens K. Nørskov Festschrift issue. The prose is readable, but the two-column layout interleaves sentences and the rate equations are garbled, so the model is described in words below.
Question
Dynamic, or programmable, catalysis oscillates adsorbate binding energies (BEs) so that surface reaction and product desorption are each favored during part of a cycle, with the aim of exceeding the Sabatier limit of a static catalyst. Earlier simulations integrated the stiff rate equations forward in time until a cyclic steady state (CSS) emerged, which becomes slow at high forcing frequencies. This paper asks how to compute the CSS efficiently and how to choose the waveform parameters (frequency, amplitude, offset, duty cycle, shape) that maximize the time-averaged turnover frequency (avTOF) (Tedesco et al. 2024).
Approach
The test system is a unimolecular reaction A to B in a well-mixed continuous stirred tank reactor, with adsorption, surface reaction and desorption steps. The forward and reverse rate constants of the surface reaction and desorption are computed from the instantaneous BE, and a site balance closes the differential-algebraic system. Two simulation strategies are compared. The sequential approach (a SciPy solveivp reimplementation of an earlier MATLAB code from the Dauenhauer group) integrates from initial conditions until the CSS is reached. The simultaneous approach, the main contribution, poses a boundary value problem in Pyomo with periodicity (limit cycle) constraints on every state variable, discretizes it with implicit Euler through Pyomo.DAE, and solves the resulting square nonlinear system with IPOPT, so only one period needs to be simulated. Waveforms studied were square, sine, triangle, sawtooth, a smoothed square wave with adjustable asymmetry and smoothness, and single or summed Gaussian pulses. Wave parameters were optimized by wrapping the simulation in scipy.optimize.minimize, using derivative-free Nelder-Mead and Powell methods for piecewise waveforms and gradient-based L-BFGS-B for continuous ones; direct optimization of continuous waveforms inside IPOPT gave matching optima. Results are interpreted with an “extended Sabatier volcano”, in which the two volcano legs are extrapolated to show the rate each promoted step could reach alone. All code is released as Colab-ready notebooks on Zenodo.
Key findings
The sequential approach reached the same CSS as the simultaneous approach, but its run times grew with forcing frequency and could be orders of magnitude longer, since more cycles must be integrated. The simultaneous approach solved a single cycle with run times nearly independent of frequency and stiffness, and its TOF trajectories satisfied the periodic boundary conditions.
The optimal duty cycle (fraction of the period in the strong-binding state) depends on the ratio of the surface reaction rate constant in the strong state to the desorption rate constant in the weak state. Systems with ratios near 0.1, 1 and 10 were studied at 1500 Hz; when surface reaction is the slower favored step, more time in the strong state is preferred. For one parameter set the optimal duty cycle was 0.69 rather than a symmetric 0.5, consistent with the TOF curve showed product desorption finishing before the wave switched. Two-variable optimization of duty cycle and offset at amplitudes of 0.2, 0.3 and 0.4 eV showed that larger amplitudes give higher rates and make the duty cycle less important. When frequency and amplitude were also optimized, both went to their upper bounds.
The best square wave (1500 Hz, amplitude 0.40 eV, offset 1.37 eV, duty cycle 0.78) gave an avTOF of 853 s-1, compared with 0.05 s-1 at the static Sabatier peak: an enhancement factor of about 1.7 x 104, and about 5.2 x 105 relative to the average of the static TOFs at the two BE states. At the optimum, desorption has the larger rate constant in its favored state, balanced by a high duty cycle; the authors link this to the steeper desorption leg and to the reverse surface reaction being favored in the weak state. A design with a flat offset line needed a larger amplitude (0.49 eV) and reached a slightly lower 845 s-1, showing that optimization beats simply raising amplitude. For a system from prior work with a steep offset line, the optimal duty cycle exceeded 0.99 and avTOF was only 0.8 s-1, suggesting narrow Gaussian pulses there. Square waves gave the highest avTOF among the shapes tested. Two summed Gaussian pulses gave 828.8 s-1 when overlapped and 806.6 s-1 when separated (1300 Hz); separated pulses win below a threshold near 1300 Hz, effectively doubling the frequency, and overlapped, square-like pulses win above it.
Significance
The work shows that a limit-cycle BVP solved in an equation-oriented modeling language makes CSS simulation of programmable catalysts fast enough to embed in optimization, and that waveform parameters beyond frequency and shape (duty cycle, offset) materially change the rate. The extended Sabatier volcano is offered as a qualitative design aid, not a replacement for optimization. The authors caution that physical stimulus limits and mass or heat transport may cap real enhancements, and that extension to multistep and multimolecular reactions remains open (Tedesco et al. 2024).